Fillet surface and offset surface

(a) Fillet surface

A fillet surface is a smooth blending surface that joins two intersecting surfaces (or a surface and a plane) with a rounded transition. It removes sharp edges.

• Constant-radius fillet: the surface is the envelope of a ball of radius $r$ rolling in contact with both surfaces; the ball centre follows the curve that lies at distance $r$ from both. The section is a circular arc tangent to both surfaces.
• Variable-radius fillet: the radius changes along the edge.
• A chamfer is the flat version.
• For a given radius, the arc length of the fillet depends on the angle at which the two faces meet.

   before            after fillet
   |                 |
   |____             |__
                        `--.____   (rounded corner)

Uses: reduces stress concentration in machine parts, makes castings and moulds easier to manufacture (draft and flow), improves appearance and safety, and avoids sharp edges in sheet-metal and plastic parts.

(b) Offset surface

An offset surface is a surface at a constant distance $d$ along the normal from a given surface:

$$\mathbf P_o(u,w)=\mathbf P(u,w)+d\,\hat{\mathbf n}(u,w)$$

Examples: offsetting a cylinder of radius 20 by $d=3$ gives radius 23 (outward) or 17 (inward); a plane offset is a parallel plane.

• Limitation: if $d$ is larger than the smallest radius of curvature on the concave side, the offset self-intersects (a cusp or loop appears).
• Uses: cutter-path generation (offset the part surface by the tool radius so the tool centre follows it), shelling and wall thickness (hollow parts, plastic housings), clearance and tolerance zones, mould cavities made from a part with shrinkage allowance, and sheet-metal thickness.